<!DOCTYPE html>
<html lang="en">
  <head>
  <meta charset="utf-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="author" content="Zhou Wei <zromyk@163.com>">
  <title>数学-高等数学 第18讲 多元函数积分学</title>
  <link rel="shortcut icon" href="/favicon.ico">
  <link rel="stylesheet" href="/style/pure.css">
  <link rel="stylesheet" href="/style/main.css">
  <link rel="stylesheet" href="https://cdn.staticfile.org/font-awesome/4.7.0/css/font-awesome.css">
  <link href="https://apps.bdimg.com/libs/highlight.js/9.1.0/styles/default.min.css" rel="stylesheet">
  <script src='/style/baidu.js'></script>
</head>
<body>
  <div id="menu-background"></div>
  <div id="menu">
    <div class="pure-menu pure-menu-horizontal">
  <div id="menu-block">
    <ul class="pure-menu-list">
      <a class="pure-menu-heading" href="/index.html">ZROMYK</a>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/index.html">主页</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/public/archive/index.html">归档</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/public/download/index.html">下载</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/public/feedback/index.html">反馈</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/public/about/index.html">关于我</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="https://github.com/zromyk"><i class="fa fa-github" style="font-size:32px"></i></a>
</li>

    </ul>
  </div>
</div>

  </div>
  <div id="layout">
    <div class="content">
      <div id="content-articles">
  <h1 id="数学-高等数学 第18讲 多元函数积分学" class="content-subhead">数学-高等数学 第18讲 多元函数积分学</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
    <span id="/public/article/数学-高等数学 9 第18讲 多元函数积分学.html" class="leancloud_visitors" style="display:none" data-flag-title="数学-高等数学 第18讲 多元函数积分学"></span>
  </p>
  <h2 id="18">第18讲 多元函数积分学</h2>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>多元函数积分学</th>
<th>表达式</th>
<th>坐标系</th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td>1.二重积分</td>
<td>
<script type="math/tex">\iint_D f(x,y)d\sigma</script>
</td>
<td>直角坐标系</td>
<td>极坐标系</td>
<td></td>
</tr>
<tr>
<td>2.三重积分</td>
<td>
<script type="math/tex">\iiint_\Omega f(x,y,z)dv</script>
</td>
<td>直角坐标系</td>
<td>柱面坐标系</td>
<td>球面坐标系</td>
</tr>
<tr>
<td>3.第一型曲线积分</td>
<td>
<script type="math/tex">\int_Lf(x,y)ds</script><br /><script type="math/tex">\int_\Gamma f(x,y,z)ds</script>
</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>4.第二型曲线积分</td>
<td>
<script type="math/tex">\int_L Pdx + Qdy</script><br /><script type="math/tex">\int_\Gamma Pdx+Qdy+Rdz</script>
</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>（1）格林公式</td>
<td></td>
<td>二重积分</td>
<td></td>
<td></td>
</tr>
<tr>
<td>（2）斯托克斯公式</td>
<td></td>
<td>二重积分</td>
<td></td>
<td></td>
</tr>
<tr>
<td>5.第一型曲面积分</td>
<td>
<script type="math/tex">\iint_\Sigma f(x,y,z)dS</script>
</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>6.第二型曲面积分</td>
<td>
<script type="math/tex">\iint_\Sigma Pdydz + Qdzdx + Rdxdy</script>
</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>（1）高斯公式</td>
<td></td>
<td>三重积分</td>
<td></td>
<td></td>
</tr>
</tbody>
</table></div>
<h3 id="1">1. 二重积分</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(直角坐标系)\iint_D f(x,y)d\sigma &= \int_{a}^{b} dx \int_{y_1(x)}^{y_2(x)} f(x,y)dy \\[1ex]
&= \int_{a}^{b} dy \int_{x_1(x)}^{x_2(x)} f(x,y)dx
\end{split}\end{equation}
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学 第18讲 多元函数积分学.assets/IMG_0092-3451157.jpg" alt="IMG_0092" style="zoom:18%;" /><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\begin{cases}
x = r\cos\theta \\[2ex]
y = r\sin\theta \\[2ex]
d\sigma = dxdy = r\ dr d\theta
\end{cases} \\[1em]
(极坐标系)(a)\iint_{D}f(x,y)d\sigma &= \int_\alpha^\beta d\theta\int_{r_1(\theta)}^{r_2(\theta)}f(r\cos\theta,r\sin\theta)rdr\text{（极点O在区域D外部）}\\
(b)\iint_{D}f(x,y)d\sigma &= \int_\alpha^\beta d\theta\int_0^{r(\theta)}f(r\cos\theta,r\sin\theta)rdr\text{（极点O在区域D边界上）} \\
(c)\iint_{D}f(x,y)d\sigma &= \int_0^{2\pi}d\theta\int_0^{r(\theta)}f(r\cos\theta,r\sin\theta)rdr\text{（极点O在区域D内部）}
\end{split}\end{equation}
</script>
</p>
<h3 id="2">2. 三重积分</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(直角坐标系)\iiint_\Omega f(x,y,z)dv &= \int_{a}^{b} dx \int_{y_1(x)}^{y_2(x)} dy \int_{z_1(x,y)}^{z_2(x,y)} f(x,y,z)dz \\[1em]
&= \int_{a}^{b} dz \iint_{D_{xy}} f(x,y,z) dxdy \\[3em]
& \begin{cases}
x = r\cos\theta \\[2ex]
y = r\sin\theta \\[2ex]
z = z \\[2ex]
dv = dxdydz = r\ \ dr d\theta dz
\end{cases} \\[1em]
(柱面坐标系) &= \iiint_\Omega f(r\cos\theta,\ r\sin\theta,\ z)r\ \ dr d\theta dz \\[3em]
& \begin{cases}
x = r\sin\varphi\cos\theta \\[2ex]
y = r\sin\varphi\sin\theta \\[2ex]
z = r\cos\varphi \\[2ex]
dv = dxdydz = r^2\sin\varphi\ \ d\theta d\varphi dr
\end{cases} \\[1em]
(球面坐标系) &= \iiint_\Omega f(r\sin\varphi\cos\theta,\ r\sin\varphi\sin\theta,\ r\cos\varphi)r^2\sin\varphi\ \ d\theta d\varphi dr
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>
<script type="math/tex; mode=display">
(2009.12)
\begin{equation}\begin{split}
设\ \Omega=\{(x,y,z)&|x^2+y^2+z^2\le1\} \\[1ex]
\iiint_\Omega z^2dv &= \cfrac{1}{3}\iiint_\Omega x^2+y^2+z^2dv=\cfrac{1}{3}\iiint_\Omega dv\ \ \ \ \  错误！！！\\[1ex]
&=\cfrac{1}{3}\iiint_\Omega r^2r^2\sin\varphi d\theta d\varphi dr \\[1ex]
&=\cfrac{1}{3}\int_0^{2\pi}d\theta\int_0^{\pi}\sin\varphi\int_0^1r^4dr \\[1ex]
&=\cfrac{1}{3}·2\pi·2·\cfrac{1}{5}=\cfrac{4\pi}{15}
\end{split}\end{equation}
</script>
</p>
</blockquote>
<h3 id="3"><a style="color:rgb(0,141,255)">3. 第一型曲线积分</a></h3>
<p>（数量值函数在曲线上的积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(二维)\int_Lf(x,y)ds &= \int_\alpha^\beta f[x(t),y(t)]\sqrt{(x_t')^2+(y_t')^2}dt \\[1ex]
&= \int_\alpha^\beta f[x,y(x)]\sqrt{1+(y_x')^2}dx \\[1ex]
&= \int_\alpha^\beta f[r(\theta)\cos\theta,r(\theta)\sin\theta]\sqrt{(r_\theta)^2+(r_\theta')^2}d\theta
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(三维)\int_\Gamma f(x,y,z)ds &= \int_\alpha^\beta f[x(t),y(t),z(t)]\sqrt{(x_t')^2+(y_t')^2+(z_t')^2}dt \\[1ex]
&= \int_\alpha^\beta f[x,y(x),z(x)]\sqrt{1+(y_x')^2+(z_x')^2}dx
\end{split}\end{equation}
</script>
</p>
<p>形心<br />
<script type="math/tex; mode=display">
(\overline{x},\overline{y},\overline{z})
=(\cfrac{\int_Lxds}{\int_Lds},\cfrac{\int_Lyds}{\int_Lds},\cfrac{\int_Lzds}{\int_Lds})
</script>
<br />
重心（质心）<br />
<script type="math/tex; mode=display">
(\overline{x},\overline{y},\overline{z})
=(\cfrac{\int_Lx\rho(x,y,z)ds}{\int_L\rho(x,y,z)ds},\cfrac{\int_Ly\rho(x,y,z)ds}{\int_L\rho(x,y,z)ds},\cfrac{\int_Lz\rho(x,y,z)ds}{\int_L\rho(x,y,z)ds})
</script>
<br />
转动惯量（对 <script type="math/tex">x,y,z</script> 轴和原点 <script type="math/tex">O</script> 的转动惯量）<br />
<script type="math/tex; mode=display">
I_x=\int_L(y^2+z^2)\rho(x,y,z)ds \\[1ex]
I_y=\int_L(z^2+x^2)\rho(x,y,z)ds \\[1ex]
I_z=\int_L(x^2+y^2)\rho(x,y,z)ds \\[1ex]
I_O=\int_L(x^2+y^2+z^2)\rho(x,y,z)ds \\
</script>
</p>
<h3 id="4"><a style="color:rgb(0,141,255)">4. 第二型曲线积分</a></h3>
<p>（向量值函数在曲线上的积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_L\overrightarrow{\pmb{F}}(x,y)\overrightarrow{\pmb{ds}} 
&= \int_L Pdx + Qdy \\[1ex]
&= \int_{t_1}^{t_2} \bigg\{\ P[x(t),y(t)]x_t'(t) + Q[x(t),y(t)]y_t'(t)\ \bigg\}dt
\end{split}\end{equation}
</script>
</p>
<h4 id="_1">【二维】第一二型曲线积分的关系</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_L Pdx + Qdy 
&= \int_{t_1}^{t_2} (P\cos\alpha + Q\cos\beta)ds \\[1ex]
&= \int_{t_1}^{t_2} (P\cos\alpha + Q\sin\alpha)ds\\[2em]
方向余弦\ \cos\alpha &= \cfrac{x_t'(t)}{\sqrt{(x_t')^2+(y_t')^2}} \\[1ex]
\cos\beta &= \cfrac{y_t'(t)}{\sqrt{(x_t')^2+(y_t')^2}} = \sin\alpha \\[2ex]
\overrightarrow{\pmb{l}}=(\cos\alpha,\cos\beta)\ &为\ L\ 上点\ (x,y)\ 处与\ L\ 同向的单位切向量. \\[1ex]
=(\cos\alpha,\sin\alpha)\ 
\end{split}\end{equation}
</script>
</p>
<h4 id="_2">【二维】格林公式</h4>
<p>（第二型曲线积分与二重积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\oint_LPdx+Qdy &= \iint_D(\cfrac{\partial Q}{\partial x} - \cfrac{\partial P}{\partial y})dxdy
\end{split}\end{equation}
</script>
</p>
<p>////////////////////////////////////////////////////////<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_\Gamma\overrightarrow{\pmb{F}}(x,y)\overrightarrow{\pmb{ds}} 
&= \int_\Gamma Pdx + Qdy +Rdz \\[1ex]
&= \int_{t_1}^{t_2} \bigg\{\ P[x(t),y(t),z(t)]x_t'(t) + Q[x(t),y(t),z(t)]y_t'(t) + R[x(t),y(t),z(t)]z_t'(t)\ \bigg\}dt
\end{split}\end{equation}
</script>
</p>
<h4 id="_3">【三维】第一二型曲线积分的关系</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\oint_\Gamma Pdx+Qdy+Rdz
&= \int_{t_1}^{t_2} (P\cos\alpha + Q\cos\beta + R\cos\gamma)ds \\[1ex]
方向余弦\ \cos\alpha &= \cfrac{x_t'(t)}{\sqrt{(x_t')^2 + (y_t')^2 + (z_t')^2}} \\[1ex]
\cos\beta &= \cfrac{y_t'(t)}{\sqrt{(x_t')^2 + (y_t')^2 + (z_t')^2}} \\[2ex]
\cos\gamma &= \cfrac{z_t'(t)}{\sqrt{(x_t')^2 + (y_t')^2 + (z_t')^2}} \\[2ex]
\overrightarrow{\pmb{l}}=(\cos\alpha,\cos\beta,\cos\gamma)\ &为\ L\ 上点\ (x,y,z)\ 处与\ L\ 同向的单位切向量.
\end{split}\end{equation}
</script>
</p>
<h4 id="_4">【三维】斯托克斯公式</h4>
<p>（第二型曲线积分与二重积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\oint_\Gamma Pdx+Qdy+Rdz
&= \iint_D \begin{vmatrix}
dydz & dzdx & dxdy \\
\cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix} &\text{(第二型曲面积分)} \\[1ex]
&= \iint_D \begin{vmatrix}
\cos\alpha & \cos\beta & \cos\gamma  \\
\cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix} dS \ \ \ \ &\text{(第一型曲面积分)}
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\oint_\Gamma Pdx+Qdy+Rdz = \iint_D
 (\cfrac{\partial R}{\partial y} - \cfrac{\partial Q}{\partial z})dydz
+(\cfrac{\partial P}{\partial z} - \cfrac{\partial R}{\partial x})dzdx
+(\cfrac{\partial Q}{\partial x} - \cfrac{\partial P}{\partial y})dxdy
</script>
</p>
<h3 id="5"><a style="color:rgb(0,141,255)">5. 第一型曲面积分</a></h3>
<p>（数量值函数在曲面上的积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iint_\Sigma f(x,y,z)dS &= \iint_{D_{xy}} f(x,y,z(x,y))\sqrt{1+(z_x')^2+(z_y')^2}dxdy
\end{split}\end{equation}
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学 第18讲 多元函数积分学.assets/IMG_0093-3451157.jpg" alt="IMG_0093" style="zoom: 16%;" /></p>
<p>形心<br />
<script type="math/tex; mode=display">
(\overline{x},\overline{y},\overline{z})
=(\cfrac{\iint_\Sigma xdS}{\iint_\Sigma dS},\cfrac{\iint_\Sigma ydS}{\iint_\Sigma dS},\cfrac{\iint_\Sigma zdS}{\iint_\Sigma dS})
</script>
<br />
曲面重心（质心）<br />
<script type="math/tex; mode=display">
(\overline{x},\overline{y},\overline{z})
=(\cfrac{\iint_\Sigma x\rho(x,y,z)dS}{\iint_\Sigma \rho(x,y,z)dS},\cfrac{\iint_\Sigma y\rho(x,y,z)dS}{\iint_\Sigma \rho(x,y,z)dS},\cfrac{\iint_\Sigma z\rho(x,y,z)dS}{\iint_\Sigma \rho(x,y,z)dS})
</script>
<br />
转动惯量（对 <script type="math/tex">x,y,z</script> 轴和原点 <script type="math/tex">O</script> 的转动惯量）<br />
<script type="math/tex; mode=display">
I_x=\iint_\Sigma(y^2+z^2)\rho(x,y,z)dS \\[1ex]
I_y=\iint_\Sigma(z^2+x^2)\rho(x,y,z)dS \\[1ex]
I_z=\iint_\Sigma(x^2+y^2)\rho(x,y,z)dS \\[1ex]
I_O=\iint_\Sigma(x^2+y^2+z^2)\rho(x,y,z)dS \\
</script>
</p>
<h3 id="6"><a style="color:rgb(0,141,255)">6. 第二型曲面积分</a></h3>
<p>（向量值函数在曲面上的积分）</p>
<p>第二型曲面积分的被积函数 <script type="math/tex">\overrightarrow{\pmb{F}}(x,y,z) = P(x,y,z)\overrightarrow{\pmb{i}}+Q(x,y,z)\overrightarrow{\pmb{j}}+R(x,y,z)\overrightarrow{\pmb{k}}</script> 定义在光滑的空间有向曲面 <script type="math/tex">\Sigma</script> 上，其物理背景是向量函数 <script type="math/tex">\overrightarrow{\pmb{F}}(x,y,z)</script> 通过曲面 <script type="math/tex">\Sigma</script> 的通量.<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iint_\Sigma\overrightarrow{\pmb{F}}(x,y,z)\overrightarrow{\pmb{dS}} 
&=\ \ \ \iint_\Sigma \ \ \ Pdydz + Qdzdx + Rdxdy \\[1ex]
&=\ \ \ \iint_{\Sigma} \ \ \ P(x(y,z),y,z)dydz + \iint_{\Sigma} \ \ \ Q(x,y(z,x),z)dzdx + \iint_{\Sigma} \ \ \ R(x,y,z(x,y))dxdy \\[1ex]
&=\pm \iint_{D_{yz}} P(x(y,z),y,z)dydz \pm\iint_{D_{zx}} Q(x,y(z,x),z)dzdx \pm\iint_{D_{xy}} R(x,y,z(x,y))dxdy
\end{split}\end{equation}
</script>
</p>
<ol>
<li>
<p>当 <script type="math/tex">\Sigma</script> 的法向量与 <script type="math/tex">x,y,z</script> 轴的夹角为 <strong>锐角</strong> 时， <script type="math/tex">\pm</script> 取 <script type="math/tex">+</script>
</p>
</li>
<li>
<p>当 <script type="math/tex">\Sigma</script> 的法向量与 <script type="math/tex">x,y,z</script> 轴的夹角为 <strong>钝角</strong> 时， <script type="math/tex">\pm</script> 取 <script type="math/tex">-</script>
</p>
</li>
<li>
<p>即 <script type="math/tex"> \Sigma </script> 前右上侧为 <script type="math/tex"> + </script> ，后左下侧为 <script type="math/tex"> - </script>
</p>
</li>
</ol>
<h4 id="_5">第一二型曲面积分的关系</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&(第一型曲面积分) \\[1ex]
\iint_\Sigma\overrightarrow{\pmb{F}}(x,y,z)\overrightarrow{\pmb{dS}}
&= \iint_\Sigma Pdydz + Qdzdx + Rdxdy\\[1ex]
&= \iint_\Sigma (P\cos\alpha + Q\cos\beta + R\cos\gamma)dS \\[1ex]
&= \iint_\Sigma (P\cfrac{\cos\alpha}{\cos\gamma} + Q\cfrac{\cos\beta}{\cos\gamma} + R)\cos\gamma \ dS \\[1ex]
&= \iint_\Sigma (P(-z_x') + Q(-z_y') + R)dS \\[3em]
方向余弦\ \cos\alpha &= \cfrac{-z_x'}{\sqrt{1+(z_x')^2+(z_y')^2}} \\[1ex]
\cos\beta &= \cfrac{-z_y'}{\sqrt{1+(z_x')^2+(z_y')^2}} \\[1ex]
\cos\gamma &= \cfrac{1}{\sqrt{1+(z_x')^2+(z_y')^2}}\end{split}\end{equation}
</script>
</p>
<h4 id="_6">高斯公式</h4>
<p>（第二型曲面积分与三重积分）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iiint_\Omega(\cfrac{\partial P}{\partial x} + \cfrac{\partial Q}{\partial y} + \cfrac{\partial R}{\partial z})dv &= \oiint_\Sigma Pdydz+Qdzdx+Rdxdy \\[1ex]
&= \oiint_\Sigma (P\cos\alpha + Q\cos\beta + R\cos\gamma)dS
\end{split}\end{equation}
</script>
</p>
<h3 id="7">7. 实际应用</h3>
<p>在求解时应该充分利用函数的对称性，如奇对称、偶对称、<u>轮换对称性</u>&hellip;</p>
</div>
<div id="nav">
  <div class="navigation">
  <ul class="pure-menu-list">
    <li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#18">第18讲 多元函数积分学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1">1. 二重积分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2">2. 三重积分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3"><a style="color:rgb(0,141,255)">3. 第一型曲线积分</a></a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#4"><a style="color:rgb(0,141,255)">4. 第二型曲线积分</a></a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_1">【二维】第一二型曲线积分的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_2">【二维】格林公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_3">【三维】第一二型曲线积分的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_4">【三维】斯托克斯公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#5"><a style="color:rgb(0,141,255)">5. 第一型曲面积分</a></a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#6"><a style="color:rgb(0,141,255)">6. 第二型曲面积分</a></a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_5">第一二型曲面积分的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_6">高斯公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#7">7. 实际应用</a>
</li>

  </ul>
</div>

</div>
 
    </div>
  </div>
  <div id="footer">
    <div class="legal pure-g">
  <div class="pure-u-1 u-sm-1-2">
    <p class="legal-license"><a href="https://beian.miit.gov.cn/#/Integrated/index">浙ICP备2020038748号</a></p>
  </div>
  <div class="pure-u-1 u-sm-1-2">
    <p class="legal-links"><a href="https://github.com/zromyk">GitHub</a></p>
    <p class="legal-copyright">Copyright © 2021 Wei Zhou. 保留所有权利。</p>
  </div>
</div>
  </div>
  <script src='/style/latest.js?config=TeX-MML-AM_CHTML'></script>
  <script src="https://cdn.bootcss.com/jquery/3.2.1/jquery.min.js"></script>
  <script src='/style/Valine.min.js'></script>
  <script src="https://apps.bdimg.com/libs/highlight.js/9.1.0/highlight.min.js"></script>
  <script type="text/javascript">
    hljs.initHighlightingOnLoad();
  </script>
  <script src="https://cdn.geogebra.org/apps/deployggb.js"></script>
  <script src="https://cdn1.lncld.net/static/js/2.5.0/av-min.js"></script>
  <script src='/style/readTimes.js'></script>
</body>
</html>
